The subject of this dissertation is to develop a rigorous transverse-mode integral equation formulation for analyzing TE/TM electromagnetic mode field solutions for dielectric waveguides. The main topics are composed of two related parts. The first part deals with scalar problems. In which we propose a transverse-mode integral-equation formulation for problems such as mode solutions of the ridged microwave waveguides. This same technique also applies to EM waves scattering off the facet of dielectric slab waveguides terminating in free space. For both problems we constructed a specifically chosen basis for the unknown tangential field functions, and we were able to reduce the kernel matrix size by more than one half without noticeable degradation of the field solutions. In the second part of the thesis, we apply a full-vector integral-equation formulation to analyze modal characteristics of the complex, two-dimensional, rectangular-like dielectric waveguide that is divisible into vertical slices of one-dimensional layered structures. The entire electromagnetic vector mode field solution is completely determined by the y-component electric and magnetic field functions on the interfaces between slices. These interfacial functions are governed by a system of vector-coupled transverse-mode integral equations (VCTMIE) whose kernels are made of orthonormal sets of both TE-to-y and TM-to-y modes from each slice. Finally, we show the numerical results to present the stable and quick convergence of this method as well as to improve the Gibb¡¦s phenomenon in the recreation of the transverse fields.